The universality of the TBA has, in fact, a geometric underpinning which follows from the geometric interpretion [1]} of the classical TBA developed in Section REF . Recall that in the geometric viewpoint, the interaction can be implemented as a density-dependent change of metric on a system of free particles. This is because the scattering shifts \(\delta (\eta ,\mu )\)  effectively change the perceived space. In this interpretation, each “mode" \(\mu \)  induces an extra amount of physical space given by \(\delta (\eta ,\mu )\)  as perceived by the mode \(\eta \) . This change of metric is exactly what we need to take into account when recasting the entropy factor in (REF ) as a constraint on the asymptotoic coordinates \(x_n^-\) .
